# 𝐴_{∞}-structures associated with pairs of 1-spherical objects and noncommutative orders over curves

@article{Polishchuk2020\_structuresAW,
title={𝐴\_\{∞\}-structures associated with pairs of 1-spherical objects and noncommutative orders over curves},
author={Alexander Polishchuk},
journal={Transactions of the American Mathematical Society},
year={2020}
}
• A. Polishchuk
• Published 29 May 2018
• Mathematics
• Transactions of the American Mathematical Society
<p>We show that pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,Y)</mml:annotation> </mml:semantics> </mml…
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## References

SHOWING 1-10 OF 49 REFERENCES

### Moduli of curves as moduli of $A_{\infty}$-structures

We define and study the stack ${\mathcal U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus g with g smooth marked points forming an ample non-special divisor. We define an

### Arithmetic mirror symmetry for genus 1 curves with n marked points

• Mathematics
• 2016
We establish a $${\mathbb {Z}}[[t_1,\ldots , t_n]]$$Z[[t1,…,tn]]-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category

### Local cohomology for non-commutative graded algebras

We generalize the theory of local cohomology and local duality to a large class of non-commutative N‐graded noetherian algebras; specifically, to any algebra, B, that can be obtained as graded

### Modular compactifications of the space of pointed elliptic curves I

• D. Smyth
• Mathematics
Compositio Mathematica
• 2010
Abstract We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford

### A modular compactification of $\mathcal{M}_{1,n}$ from $A_\infty$-structures

• Mathematics
• 2014
We show that a certain moduli space of minimal $A_\infty$-structures coincides with the modular compactification $\bar{\mathcal{M}}_{1,n}(n-1)$ of $\mathcal{M}_{1,n}$ constructed by Smyth. In

### MODULI OF CURVES WITH NONSPECIAL DIVISORS AND RELATIVE MODULI OF $A_{\infty }$ -STRUCTURES

• A. Polishchuk
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2017
In this paper, for each $n\geqslant g\geqslant 0$ we consider the moduli stack $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ of curves $(C,p_{1},\ldots ,p_{n},v_{1},\ldots ,v_{n})$ of arithmetic genus $g$

### A modular compactification of ℳ1,n from A ∞-structures

• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2017
Abstract We show that a certain moduli space of minimal A ∞ A_{\infty} -structures coincides with the modular compactification ℳ ¯ 1 , n ⁢ ( n - 1 ) {\overline{\mathcal{M}}}_{1,n}(n-1) of ℳ 1 , n

### Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry

• Mathematics
• 2008
We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies several

### Extensions of homogeneous coordinate rings to $A_ \infty$-algebras

We study $A_{\infty}$-structures extending the natural algebra structure on the cohomology of $\oplus_n L^n$, where $L$ is a very ample line bundle on a projective $d$-dimensional variety $X$ such

### Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the Yang–Baxter Equations

We show that a nondegenerate unitary solution r(u, v) of the associative Yang–Baxter equation (AYBE) for $$\mathrm{Mat}(N,\mathcal{C})$$ (see [7]) with the Laurent series at u = 0 of the form r(u, v)